Le spectre résiduel de GL(n). (The residual spectrum for GL(n)). (French) Zbl 0696.10023
If G is a reductive algebraic group defined over a global field k then Langlands showed how the \(G(k_ A)\)-representation on \(L^ 2(G(k)\setminus G(k_ A))\) can be decomposed into components corresponding to cuspidal representations of Levi components of parabolic subgroups up to a certain equivalence relation. The discrete part of \(L^ 2(G(k)\setminus G(k_ A))\) has a corresponding decomposition and the components are generated by the residues of Eisenstein series. In the case \(G=GL_ 2\) the only discrete and non-cuspidal components are 1- dimensional. In this remarkable paper the authors give a wide-ranging generalization of this fact for \(GL_ N.\)
Indeed they classify all the discrete components in this case. The proof is long and intricate. The local part depends on a close study of the intertwining operators and makes particular use of Zelevinski’s classification of representations of \(GL_ N\). The global part makes use of the theory of L-functions for \(GL_ N\) developed by Jacquet, Piatetski-Shapiro and Shalika.
Indeed they classify all the discrete components in this case. The proof is long and intricate. The local part depends on a close study of the intertwining operators and makes particular use of Zelevinski’s classification of representations of \(GL_ N\). The global part makes use of the theory of L-functions for \(GL_ N\) developed by Jacquet, Piatetski-Shapiro and Shalika.
Reviewer: S.J.Patterson
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 14G25 | Global ground fields in algebraic geometry |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
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